Simplifications and clarifications on the paper 'An algebra of transfer functions for distributed linear time-invariant systems'

In this note we first point out some simplifications in some results of our paper mentioned above [1]. Second, we prove that the algebra of transfer functions
\hat{\cal B}(\sigma_0)
, introduced in the paper, is in fact the quotient ring of the ring
\hat{\cal Q}_{_}(\sigma_0)
with respect to the multiplicative system
\hat{\cal Q}_{_}^{\infty} (\sigma_0)
defined in this note. The analogy between
\hat{\cal B}(\sigma_0)
, seen as the quotient
[\hat{\cal Q}_{_}(\sigma_0)][\hat{\cal Q}_{_}^{\infty} (\sigma_0)]^{-1}
, and the algebra of proper rational functions
C_p (s)
seen as the quotient
[\Re (\sigma_0)][\Re^{\infty} (\sigma_0)]^{-1}
(where
\Re_0 (\sigma_0 )
is the ring of proper rational functions analytic in
Re s \geq \sigma_0
, and
\Re^{\infty} (\sigma_0)
is the multiplicative system of such functions tending to a nonzero constant as
|s| \irightarrow \infty
), is fully developed and supports the claim that
\hat{\cal B} (\sigma_0 )
is a natural extension of the algebra of proper rational functions to distributed systems. These algebraic developments have been found most useful in applications [11], [12].